Essential selfadjointness of the Laplacian of Riemannian manifolds and Lioville property

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03290
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Tohoku University (2022-2023); Hokkaido University (2018-2021)
• Principal Investigator: Masamune Jun 東北大学, 理学研究科, 教授 (50706538)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: ラプラシアン / 本質的自己共役性 / 保存則 / リュービル性

Outline of Final Research Achievements

In collaboration with Schmidt, we defined the conservation law of heat for the Schrodinger operator and obtained necessary and sufficient conditions for the Kasiminski-type to be satisfied. This result was published in Math Ann. In collaboration with Hua and Wojciechowski, we clarified the relation between essential self adjointness and L^2 Liouville property for the Laplacian on continua and discrete graphs, and the results were published by JFAA. In collaboration with Hinz and Suzuki, it was shown that for a noncomplete Riemannian manifold obtained by removing compact and closed sets from a complete space, a necessary and sufficient condition for essential self adjointness is that the Cauchy boundary is polar at some appropriate capacity. These results have been published in Non Linear Analysis. In collaboration with Inoue, Ku, and Wojciechowski, we gave another proof of the classical Hamburger's theorem for the Laplacian over natural numbers. We submitted this result to a journal.

Free Research Field

大域解析学

Academic Significance and Societal Importance of the Research Achievements

ラプラシアンの本質的自己共役性は対応するダイナミックスの境界や特異集合の付近での振る舞いが決定されることと同値であるため，解析学や幾何学における古くから研究をされている基本的な問題であるが，未だ分かっていないことが多く，とりわけ，空間が非完備な場合には一般的な判断基準が存在しなかった．本研究課題ではこの問題に対して出来るだけ一般的な状況で「コーシー境界が極」であることと，本質的自己共役性の関係を調べることで迫った．連続体や離散空間を調べた結果，完全な回答を得られたわけではないが，今回調べた全てのケースにおいては，これらの概念は同地であることが明らかにされた．